On the diophantine equation $D₁x² + D₂ = 2^{n+2}$

On the diophantine equation $(x^3-1)/(x-1)=(y^n-1)/(y-1)$

“…So we have t 1 ≥ 17. Then, by (8) and (36) again, we get 390625 ≤ p 8Z1 ≤ p Z1(t1−1)/2 < t 2 < 2 + 2563.42 1 + 10.98π log 5 1/2 < 112451, a contradiction. All cases have been considered, the proof is complete.…”

“…In [1] and [2], Apéry proved that N (1, D 2 , p) ≤ 2 except for N (1, 7, 2) = 5. In [7] and [8], the author proved that…”

“…The lemma is proved. (2,7,9), (3,8,11), (3,10,13), (5,14,19), (6,17,23), (6,19,25), (7,20,27), (7,22,29), (2,23,31), (10,31,41), (11,32,43), (3,35,47), (3,37,49), (13,40,53) …”

“…However, by [8], then (3) has exactly two solutions (r, s) = (1, k + 2) and (2 k+1 + 1, 3k + 2). Therefore, (69) If y ≡ 1 (mod 4) and y > 5, then we have…”

On the Diophantine System $x^2 - Dy^2 = 1-D$ and $x=2z^2-1$

“…Under the assumption that D − 1 is an odd prime power, using our theorem and some known results of quartic diophantine equations (see [1], [2], [3], [6], [7], [8], [11]), we can find all solutions of (1) with ease. As an example, we prove the following corollary.…”